Ergodic Ramsey theory

Ergodic Ramsey theory is a branch of mathematics where problems motivated by additive combinatorics are proven using ergodic theory.

History

Ergodic Ramsey theory arose shortly after Endre Szemerédi's proof that a set of positive upper density contains arbitrarily long arithmetic progressions, when Hillel Furstenberg gave a new proof of this theorem using ergodic theory. It has since produced combinatorial results, some of which have yet to be obtained by other means, and has also given a deeper understanding of the structure of measure-preserving dynamical systems.

Szemerédi's theorem

Main article: Szemerédi's theorem

Szemerédi's theorem is a result in arithmetic combinatorics, concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured^[1] that every set of integers A with positive natural density contains a k term arithmetic progression for every k. This conjecture, which became Szemerédi's theorem, generalizes the statement of van der Waerden's theorem. Hillel Furstenberg proved the theorem using ergodic principles in 1977.^[2]

See also

• IP set
• Piecewise syndetic set
• Ramsey theory
• Syndetic set
• Thick set

References

• Ergodic Methods in Additive Combinatorics
• Vitaly Bergelson (1996) Ergodic Ramsey Theory -an update
• Randall McCutcheon (1999). Elemental Methods in Ergodic Ramsey Theory. Springer. ISBN 978-3540668091.

Sources

1. Erdős, Paul; Turán, Paul (1936), "On some sequences of integers" (PDF), Journal of the London Mathematical Society, 11 (4): 261–264, CiteSeerX 10.1.1.101.8225, doi:10.1112/jlms/s1-11.4.261.
2. Furstenberg, Hillel (1977), "Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions", Journal d'Analyse Mathématique, 31: 204–256, doi:10.1007/BF02813304, MR 0498471.